The Nearest Hermitian Inverse Eigenvalue Problem Solution with Respect to the 2-Norm
This work provides a theoretical solution with proofs for a specific inverse eigenvalue problem, but the impact is incremental as it addresses a niche mathematical problem.
The paper solves the nearest Hermitian inverse eigenvalue problem in the 2-norm, providing a method to find a Hermitian matrix with given eigenvalues closest to a given matrix. Tests on random matrices and grayscale images demonstrate smoothing properties of eigenvalue corrections.
Assume that the eigenvalues of a finite hermitian linear operator have been deduced accurately but the linear operator itself could not be determined with precision. Given a set of eigenvalues $λ$ and a hermitian matrix $M$, this paper will explain, with proofs, how to find a hermitian matrix $A$ with the desired eigenvalues $λ$ that is as close as possible to the given operator $M$ according to the operator 2-norm metric. Furthermore the effects of this solution are put to a test using random matrices and grayscale images which evidently show the smoothing property of eigenvalue corrections.